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Full text of "Treatise On Analysis Vol-Ii"

72       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

set of all functions/e E such that JJ |/(OI1/2 d* ^ !/ Show that the sets Vn form a
fundamental system of neighborhoods of 0 for a metrizable topology on E compatible
with the vector space structure, that the sets Vrt are bounded with respect to this
topology, but that the convex hull of each VB is the whole space E. (Observe that every
function /e E can be written as /= J(# + h) with g,heE and

1

^2

Deduce that every continuous seminorm on E is identically zero. In particular, if a
linear form on E is continuous, it is zero.

15.    (a)   Let E be a topological real vector space, H a nonempty convex open subset of E,
and/a convex function (Section 8.5, Problem 8) defined on H. Show that/is contin-
uous on H if and only if there exists a nonempty open set U <= H on which / is
bounded above. In particular, every upper semicontinuous convex function on H is
continuous on H.

(b) Give an example of a convex function p, which is defined on a compact convex
set K c R2s lower semicontinuous and bounded in K, but not continuous at a frontier
point of K. (Take K to be a closed disk with 0 as a frontier point, and apply the
definition of Problem 12(a).) Hence give an example of a compact convex set A ^ K
such that/? can be extended to a convex function which is lower semicontinuous on A,
but not bounded on A.

16.    Let E be a Hausdorff locally convex space, X a locally compact separable metric
space, d the distance on X, A a closed subset of X, and/a continuous mapping of A
into E.

(a)    Show that there exists a denumerable covering (Un) of X  A, consisting of open
sets Un <= X  A, which is locally finite and such that, for every frontier point a of A
and every neighborhood V of a in X, there exists a neighborhood V <= V of a with
the property that each Un which meets V is contained in V. (For each x e X  A,
consider the open ball with center x and radius $d(x, A), and use (12.6.1),)

(b)    Let (hn) be a continuous partition of unity on X  A subordinate to the covering
(Un) (12.6.3). For each n let bn e Un and an e A be such that d(bn, an) g 2d(bn, A).
Let g(x) =/(#) if x e A, and

if x e X  A. Show that g is a continuous mapping of X into E which extends f and
is such that g(X) is contained in the convex hull (Problem 13) of/(A).

17. Let E, F be two locally convex spaces, / a continuous linear mapping of E into F,
and L a locally compact separable metrizable subspace of F such that /(E) n L is
dense in L. For each continuous seminorm p on E and each e > 0, show that there
exists a continuous mapping s: L - E such that p(z f(s(z))) : e for all z e L. (Con-
sider a locally finite denumerable open covering (Un) of L such that p(x'  x) g; Je
whenever x and x' lie in the same Un, a continuous partition of unity (hn) sub-
ordinate to (Un), and for each n a point an e Un n/(E). If b ef~1(an)t show that the
function s defined by s(z)  ]T bnhn(z) satisfies the required conditions.)and the Aa are real numbers ^0 such that  Aa  1.